DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/differential_equations_2.md

@@ -491,6 +491,29 @@ coefficient,
 
 $$c_n:= \int^{L}_{0} dx sin(\frac{n \pi x}{L}) \psi_{0}(x). $$
     
-
+## General recipie for seperable PDEs ##
+
+1. Make the separation ansatz to obtain separate ordinary differential 
+    equations.
+2. Choose which euation to treat as the eigenvalue equation. This will depend 
+    upon the boundary conditions. Additionally, verify that the linear 
+    differential operator $L$ in the eigenvalue equation is hermitian. 
+3.Solve the eigenvalue equation. Substitute the eigenvalues into the other 
+    equations and solve those too. 
+4. Use the orthonormal basis functions to write down the solution corresponding 
+    to the specified initial and boundary conditions. 
+
+One natural question is what if the operator $L$ from setp 2 is not hermitian? 
+It is possible to try and make it hermitian by working on a Hilbert space 
+equipped with a different inner product. This means one can consider 
+modifications to the definition of $\langle \cdot, \cdot \rangle$ such that $L$
+is hermitian with respect to the modified inner product. This type of technique 
+falls under the umbrella of *Sturm-Liouville Theory*, which forms the foundation
+for a lot of the analysis that can be done analytically on PDEs. 
+
+Another question is of course what if the equation is not separable? One 
+possible approach is to try working in a different coordinate system. There are 
+a few more analytic techniques available, however in many situations it becomes
+necessary to work with numerical methods of solution. 
 
     
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